Infinity Has Its ‘Limits’

Commenter Draracle challenged me to answer the Kalam argument: which opposes the concept of an infinite universe without “first cause.” To do so, we need to talk about math.

Infinity is a theoretical abstraction which makes calculus possible. Calculus is the basis for descriptions and predictions of behavior of objects in the physical world, (as differential equations). The same equations define a mass-spring system, an electrical circuit, processes of oscillation in chemical reactions or bacterial populations.

The analysis of all of these types of systems depends on the concept of an infinitesimal distance between points or samples–as the basis for a differential, or integral.

A circle is a polygon with an infinite number of sides. Area under a curve is the result of adding up the areas of an infinite number of infinitely narrow rectangles. You can’t do this in fact, but you can do it with an integral. Slope of a curve is determined as the slope of a line between two infinitely close points. You can’t actually create two infinitely close points, but you can do it with a derivative.

Update: If you want to nerd out to the math of how the concept of infinity applies to polygons, I’ve embedded this video.

Update: A video explaining the hotel paradox.

The important thing is, the concept of infinity does exist. The hotel paradox, and other counterexamples are what happens when people mistakenly apply integer-based logic to the concept of infinity.

Likewise, discussions of a ‘first cause’ are attempt to define what the rules of a universe might be, when it was in a vastly different configuration than it is today. There may have been a great deal which took place before or outside of this expanding bubble which was created by the Big Bang.

But for us, who exist in this particular causal universe, it is for all intents and purposes an impenetrable event horizon. The universe may encompass all that exists, or it may be part of a multi-verse, which may have existed forever. It’s completely beyond the scope of our current science to speculate.

Logic fails, too. And attempts to describe what happened “before” the Big Bang or what kind of cause was needed to set it in motion are as futile as trying to describe infinity with the ‘hotel’ paradox. Part of the difficulty is that space and time are related. So there might not be anything valid about the concept of “before” the Big Bang. To resolve these paradoxes, we’ll need to wait for future cosmological discoveries. We cannot demand premature closure or draw conclusions from the incomplete information now available to us.

In order to discuss this at all, people must have the capacity to grasp certain abstractions.

Draracle responded:

Yes, infinity is a tool of math, I acknowledge that. But all that this math tells us is that IF such and such approaches infinity such will happen. I don’t know if I have heard a calculus equation stating WHEN such and such reached infinity. I could be wrong, I didn’t study much calculus.

Yes, you need to know calculus to understand what I’m saying. Limit equations describe what happens as you approach infinity. Derivatives and integrals describe what happens when you get there. That’s why I said you have to have the capacity to grasp the abstraction. More people should study math. It would be a better world.

For example, if you want to calculate pi to an infinite number of decimal places, it is the result of an infinite series of numbers: You add a small amount, subtract a smaller amount, add a smaller amount, subtract a smaller amount, and so on. With each term, you get closer and closer to the absolute value of pi, but you never get there. But we can say that pi describes an actual ratio between the diameter and circumference of a circle. You can never get the exact ratio, because it requires an infinite number of terms. But this is still how the ratio is defined.

As for your circle, I am pretty sure that your description of a circle is, in fact, a dot… at best. “Slope of a curve is determined as the slope of a line between two infinitely close points” — how long is a line between two infinitely close points? What is that distance? It isn’t zero. It isn’t infinity. It isn’t negative infinity. I would argue that to infinitely close dots is rather the distance between to dots as they approach zero distance. As the approach zero distance. As soon as you define two points the distance is finite. If you continue to approach zero and define a closer spot, the distance is finite. You can reach zero. You can reach small, finite, points. But you can never reach an infinitely small distance. It is a purely theoretical tool to describe a very finite actuality.

No. Start with a triangle. Add more sides until you get to, say, an octagon. It looks more like a circle than a triangle. Now add more sides until you have 100 or 1000. At 1000, you have what amounts to a circle, unless you get really really close to it. But if you draw an actual geometrically perfect circle, you have a polygon with an infinite number of sides.

Everything in the universe is based on waves (wave-particle duality). All waves are made up of varying amounts of sine waves, (when broken down by Fourier analysis–the sinusoid being the same function that describes a circle). Side note, this is what makes the data compression in .mp3’s possible.

Calculus doesn’t work without the concept of infinity. Without it, we couldn’t know anything much about finite matter. When I say “you have to grasp the abstraction”, I’m talking about how infinity applies to finite objects and waves, and helps us analyze them. I’m certainly not talking about god.

The slope of a curve can be calculated for a point. The point, in calculus, when you take the derivative, is the same as two infinitely close points. But it still has a slope. This is the essence of the concept of the derivative.

For example: The derivative of the velocity of an object is the acceleration. The second derivative of the velocity is the rate of change of acceleration. Without recognizing the property of infinity, we could never analyze dynamic systems.

“In order to discuss this at all. people must also be willing to come to grips with certain abstractions.”

Certainly you don’t mean God. I do assume you consider god an abstraction. What abstractions do we include? What do we rule out? Do we just include the abstractions which formulate the world the way in which we would like it to be?

The idea of God is not an abstraction. It’s a hypothesis. The abstraction I’m talking about in this article is infinity, which is categorically different from a very large number. A very large number behaves more like infinity, than a very small number does. But infinity isn’t a number at all.

As for the God hypothesis, there may in fact be a creator, or more likely, an impersonal creative force or agent yet to be discovered. We don’t know what that could be, and we can’t say right now. Even if we did know, we’d be forced to ask what created God, or set that creative force into motion. And that leads us to infinite regress. Every creator would have to have a creator. And that would tend to invalidate the idea of a “first cause” at all. Finding any “first cause” only moves the problem backward, to the previous cause. So “first cause” is an oxymoron. It will break your brain if you think about it too much. Kind of like a human divide-by-zero error. Theologians solve this by claiming that God is outside space and time and therefore doesn’t need a cause. A fine bit of dualist sophistry.

So therefore all discussion on all subjects, even history, should be halted until such time that we know everything. I hear the defeatist voice of Hume again.

No. I’m not saying don’t discuss it. I’m saying don’t draw conclusions about any “first cause.” Especially, I’m saying don’t take the double-talk of theologians as explanatory.

In the words of Richard Dawkins quoted in Time magazine speaking to Francis Collins:

“My mind is not closed, as you have occasionally suggested, Francis. My mind is open to the most wonderful range of future possibilities, which I cannot even dream about, nor can you, nor can anybody else. What I am skeptical about is the idea that whatever wonderful revelation does come in the science of the future, it will turn out to be one of the particular historical religions that people happen to have dreamed up. When we started out and we were talking about the origins of the universe and the physical constants, I provided what I thought were cogent arguments against a supernatural intelligent designer. But it does seem to me to be a worthy idea. Refutable–but nevertheless grand and big enough to be worthy of respect. I don’t see the Olympian gods or Jesus coming down and dying on the Cross as worthy of that grandeur. They strike me as parochial. If there is a God, it’s going to be a whole lot bigger and a whole lot more incomprehensible than anything that any theologian of any religion has ever proposed.”